Representation of Lie Groups and Special Functions

Representation of Lie Groups and Special Functions

Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Volume 316

Representation of Lie Groups and Special Functions Recent Advances by N. Ja. Vilenkin t formerly of The Correspondence Pedagogical Institute, Moscow, Russia and A.U. Klimyk Institute for Theoretical Physics, Ukrainian Academy of Sciences, Kiev, Ukraine SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A c.i.p. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-4486-0 DOI 10.1007/978-94-017-2885-0 ISBN 978-94-017-2885-0 (ebook) The manuscript was translated from Russian by V.A. Groza and A.A. Groza Printed on acid-free paper All Rights Reserved 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of Contents Preface..... Chapter 1: h-harmonic Polynomials, h-hankel Transform, and Coxeter Groups 1.1. Coxeter Groups................ 1.1.1. Dihedral groups.... 1.1.2. Generating elements and defining relations. 1.1.3. Coxeter groups........... 1.1.4. Coxeter matrices. The classification of irreducible Coxeter groups 1.1.5. Invariants of Coxeter groups 1.1.6. Invariant bilinear forms.. 1.1.7. Irreducible representations 1.1.8. Representations on polynomials 1.1.9. Representations on a group algebra 1.1.10. Polynomials pg(t).... 1.2. The h-laplacian and h-harmonic Polynomials 1.2.1. The h-laplacian........ 1.2.2. h-harmonic polynomials 1.2.3. Differential-difference operators Ti 1.2.4. The operators T;*... 1.2.5. Averaging operator....... 1.2.6. The minimum principle..... 1.2.7. Polynomials related to representations 1.2.8. Examples of h-harmonic polynomials. 1.3. The Poisson Kernel for h-harmonic Functions 1.3.1. h-exact I-forms 1.4. 1.3.2. 1.3.3. 1.3.4. 1.3.5. 1.3.6. 1.3.7. 1.3.8. The intertwining operator Kernels Kr(x,y) The space,c2(jrn, h 2 dfl). The bilinear form on polynomials The operator exp (-6.h/2) Properties of Kr(x,y) and K(x,y) The Poisson kernel h-hankel Transform.......... 1.4.1. Definition... 1.4.2. Restriction of h-hankel transform onto the sphere X111 1 1 1 2 3 5 7 9 11 14 14 16 18 18 21 25 27 30 32 34 38 44 44 47 49 51 53 54 57 58 61 61 63 v

vi Table of Contents 1.4.3. h-bessel functions 64 1.4.4. h-hankel transform and classical special functions 64 Chapter 2: Symmetric Polynomials and Symmetric Functions 67 2.1. Simplest Symmetric Polynomials and Symmetric Functions 67 2.1.1. Partitions and their orderings......... 67 2.1.2. The ring of symmetric functions. 2.1.3. 2.1.4. 2.1.5. 2.1.6. Monomial symmetric polynomials and functions 69 Elementary symmetric functions 72 Complete symmetric functions. 73 Power-sum symmetric functions 74 Schur functions.... 77 2.2. The Scalar Product on A and Skew Schur Functions 2.2.1. The scalar product on A 2.2.2. Matrices of transitions 2.2.3. Skew Schur functions. 2.2.4 Summation formulas containing Schur functions 2.3. Hall-Littlewood Polynomials and Functions 2.3.1. Definition... 2.3.2. 2.3.3. 2.3.4. The functions q>. and S >. The scalar product on A(Q(t)) Skew Hall-Littlewood polynomials 2.4. Jack Symmetric Polynomials and Functions 2.4.1. Definition... 2.4.2. 2.4.3. Symmetric functions In(Xj a) Differential operator D( a).. 2.4.4. Duality relation.... 2.4.5. Skew Jack symmetric functions 2.4.6. Expression for J,.. in terms of J,..-1 2.4.7. 2.4.8. 2.4.9. 2.4.10. Expression for J>.(I,...,lj a) Expressions for c>.(a), r>.(a), and j>.(a) Expression for J>./,........... Jack polynomiais and zonal polynomials 80 80 83 86 89 92 92 95 96 103 105 105 109 112 115 117 119 121 123 126 126 2.5 Generalized Binomial Coefficients and Jack Polynomials 127 2.5.1. Generalized binomial coefficients....... 127 2.5.2. 2.5.3. 2.5.4. 2.5.5. The main theorem Expressions for generalized binomial coefficients Special cases of generalized binomial coefficients Relations for Jack polynomials........ 131 135 137 137

Table of Contents vii 2.5.6. 2.5.7. Estimate of Jack polynomials.... Jack polynomials of two variables.... 2.6. Macdonald Symmetric Polynomials and Functions 2.6.1. The space A(IF).... 2.6.2. The operator D.... 2.6.3. Macdonald symmetric functions and polynomials 2.6.4. Duality relation.... 2.6.5. Skew Macdonald symmetric functions 139 141 143 143 145 148 150 152 2.7. Macdonald's Orthogonal Polynomials Associated with Root Systems.... 154 2.7.1. Root systems.... 154 2.7.2. Classification of irreducible root systems 157 2.7.3. Admissible pairs of irreducible root systems 164 2.7.4. The group algebra A 166 2.7.5. Scalar products on A.... 168 2.7.6. The operator E.... 172 2.7.7. Orthogonal polynomials associated with root systems 178 2.7.8. Special cases of polynomials P>..... 179 Chapter 3: Hypergeometric Functions Related I to Jack Polynomials 185 3.1. Hypergeometric Functions Related to Jack Polynomials 185 3.1.1. Definition... 185 3.1.2. Differential equations for 2Ffd) 187 3.1.3. Integral representation of 2Ffd) 192 3.1.4. The integral relation for Jack polynomials 196 3.1.5. Properties of hypergeometric functions 197 3.1.6. Symmetric orthogonal polynomials associated to Jack polynomials... 200 3.2. Hypergeometric Functions of Two Variables 206 3.2.1. Expressions in terms of the functions IFI and 2Ft 206 3.2.2. The Appell function F4..... 209 3.2.3. Expression for 2F?) in terms of F4... 211 3.2.4. Generalized Laplace transform..... 213 3.2.5. Generalized Laguerre polynomials related to Jack polynomials... 216 3.2.6. Hankel transform................ 221 3.3. Hypergeometric Functions Associated to Root Systems 222 3.3.1. Introduction... 222

viii Table of Contents 3.3.2. 3.3.3. 3.3.4. 3.3.5. 3.3.6. 3.3.7. Zonal spherical functions 224 Hypergeometric functions associated to root systems.. 227 Symmetric Jacobi polynomials associated to root systems 237 Relations between Jack polynomials and Jacobi polynomials associated to the root system A n - 1 239 Jacobi polynomials and hypergeometric functions associated to the root system BCn...... 245 Relation between Jacobi polynomials associated to Jack polynomials and Jacobi polynomials associated to the root system BC n........ 250 3.4. Basic Hypergeometric Functions Related to Schur Polynomials............. 3.4.1. Definition... 3.4.2. Expressions for the Vandermonde determinant 3.4.3. Determinental formulas for rtp~~l 3.4.4. 3.4.5. 3.4.6. and rcp~l)... Summation formulas Integral representation Transformation properties of 2cp~1) Chapter 4: Clebsch-Gordan Coefficients and Racah Coefficients of Finite Dimensional Representations 4.1. Finite Dimensional Representations of Semisimple Lie Groups and Algebras.... 4.1.1. Semisimple Lie groups and algebras 4.1.2. 4.1.3. Finite dimensional representations Finite dimensional representations of semisimple Lie algebras..... 4.1.4. Properties of a Weyl group 252 252 254 256 259 261 263 265 265 265 269 270 273 4.2. Tensor products of finite dimensional representations 276 4.2.1. Expressions for representation multiplicities in terms of weight multiplicities... 276 4.2.2. 278 4.2.3. 4.2.4 4.2.5 4.2.6. Formulas for decomposition of tensor products Ranges of disposition of highest weights in decompositions of tensor products Upper bound for multiplicities of representations in tensor products......... The theorem on shifts of highest weights Expressions for ni 280 282 283 288

Table of Contents ix 4.3. Clebsch-Gordan Coefficients of Compact Groups 289 4.3.1. Definition... 289 4.3.2. CGC's and matrix elements of representations 291 4.3.3. Problems of uniqueness for CGC's..... 293 4.3.4. Permutation symmetry of CGC's... 296 4.4. Clebsch-Gordan Coefficients and Scalar Factors 299 4.4.1. Subgroup chains and corresponding orthonormal bases 300 4.4.2. Definition of scalar factors....... 301 4.4.3. Orthogonality relations for scalar factors 305 4.4.4. Permutation symmetries of scalar factors 306 4.5. Racah Coefficients..... 309 4.5.1. Definition... 309 4.5.2. 4.5.3. 4.5.4. 4.5.5. Special cases of RC's Permutation symmetries RC's and characters of representations The addition theorem and the Biedenharn-Elliott 312 313 315 identity 316 Chapter 5: Clebsch-Gordan Coefficients of the group U( n) and Related Generalizations of Hypergeometric Functions 317 5.1. Clebsch-Gordan Coefficients of the Group U(n) and the Denominator Function... 317 5.1.1. CGC's of the tensor product Tm T(p,o) 317 5.1.2. CGC's with multiplicities....... 319 5.1.3. CGC's with multiplicities and scalar factors 323 5.1.4. The denominator function........ 326 5.1.5. Another definition of the denominator function 328 5.1.6. The path sum formula.... 330 5.2. The algebra of Boson Operators and Clebsch-Gordan Coefficients of the Group U( n)... 333 5.2.1. Creation and annihilation operators... 333 5.2.2. The algebra of creation and annihilation operators 336 5.2.3. Boson and dual boson polynomials 338 5.2.4. Properties of boson polynomials 342 5.2.5. Construction of boson polynomials 343 5.2.6. Symmetry relation for scalar factors of the tensor product Tm T(p,o)... 349 5.2.7. Matrix elements of the operator Tm(gn-l (11'/2)) 350 5.2.8. RC's and scalar factors....... 353

x Table of Contents 5.3. Hypergeometric Series Well-Poised in U(n)........ 359 5.3.1. Generalized hypergeometric series related to the group U(n)........... 359 5.3.2. Summation formulas for well-poised series 361 5.3.3. An analogue of the Whipple formula.. 366 5.3.4. Corollaries of the generalized Whipple identity 369 5.3.5. The recurrence relation for WJn)(z). 370 5.3.6. Integral relations for F(n)......... 371 5.4. Polynomials Related to Hypergeometric Series Well-Poised in U(n)........ 373 5.4.1. Functions G~n).... 373 5.4.2. Symmetries of functions G~n). 376 5.4.3. The functions,.g~n). 378 5.4.4. The functions G~n)...... 381 5.4.5. The functions ;;'G~n)..... 383 5.5. Basic Hypergeometric Series Well-Poised in U(n) and Their Properties... 385 5.5.1. Basic hypergeometric functions well-poised in U(n) 385 5.5.2. Summation formulas... 388 5.5.3. q-analogue of the generalized Whipple formula 391 Chapter 6: Gel'fand Hypergeometric Functions 393 6.1. General Hypergeometric Series 393 6.1.1. Introduction... 393 6.1.2. Horn hypergeometric series 395 6.1.3. Gel'fand general hypergeometric series 398 6.1.4. 6.1.5. General hypergeometric series associated with subspaces 399 General hypergeometric series with common convergence domain... 6.2. Gel'fand General Hypergeometric Functions 6.2.1. General hypergeometric systems of equations 6.2.2. Spaces of general hypergeometric functions 6.2.3. General hypergeometric functions associated with subspaces......... 6.2.4. Generalized hypergeometric functions 6.3. Gel'fand q-hypergeometric Series and (~, D)-Hypergeometric Series 6.3.1. Horn q-hypergeometric series 402 404 404 406 407 410 412 412

Table of Contents xi 6.3.2. General q-hypergeometric series....... 413 6.3.3. (V', D)-Hypergeometric series... 415 6.3.4. Difference analogues of hypergeometric functions 417 6.4. Hypergeometric Functions on Real Grassmannians 419 6.4.1. Real Grassmannians... 419 6.4.2. The Radon transform............ 420 6.4.3. Hypergeometric functions on Grassmannians 423 6.4.4. Hypergeometric systems of equations on Grassmannian 424 6.5. Hypergeometric Functions and Hypergeometric Series on Complex Grassmannians.............. 428 6.5.1. Hypergeometric systems of equations and hypergeometric functions on V.... 428 6.5.2. General hypergeometric functions on G3,6(C) 432 6.5.3. General hypergeometric series on Zkn(C) 436 6.5.4. Reduction relations....... 438 6.5.5. Hypergeometric functions on strata 442 6.6. Hypergeometric Functions on Strata of Grassmannian G3,6(C)...,.... 444 6.6.1. Strata of Grassmannian 03,6(C) 444 6.6.2. General hypergeometric functions in neighborhoods of one-orbit strata 446 6.6.3. Bases of spaces of hypergeometric functions on nondegenerate strata.......... 448 6.6.4. Hypergeometric functions on strata of type A 449 6.6.5. Hypergeometric functions on strata of type B 453 6.6.6. Hypergeometric functions on Grassmannian G2,4(C) 459 6.6.7. Hypergeometric functions on strata of type C 459 Bibliography... 463 Supplementary Bibliography 484 Bibliography Notes 488 Subject Index 494

Preface In 1991-1993 our three-volume book "Representation of Lie Groups and Special Functions" was published. When we started to write that book (in 1983), editors of "Kluwer Academic Publishers" expressed their wish for the book to be of encyclopaedic type on the subject. Interrelations between representations of Lie groups and special functions are very wide. This width can be explained by existence of different types of Lie groups and by richness of the theory of their representations. This is why the book, mentioned above, spread to three big volumes. Influence of representations of Lie groups and Lie algebras upon the theory of special functions is lasting. This theory is developing further and methods of the representation theory are of great importance in this development. When the book "Representation of Lie Groups and Special Functions",vol. 1-3, was under preparation, new directions of the theory of special functions, connected with group representations, appeared. New important results were discovered in the traditional directions. This impelled us to write a continuation of our three-volume book on relationship between representations and special functions. The result of our further work is the present book. The three-volume book, published before, was devoted mainly to studying classical special functions and orthogonal polynomials by means of matrix elements, Clebsch-Gordan and Racah coefficients of group representations and to generalizations of classical special functions that were dictated by matrix elements of representations. Namely, the first volume describes interrelations of the theory of classical special functions with representations of groups of second order matrices (the groups SU(2), SU(I, 1), SL(2, R.), 1S0(2) and others). This material led to a large number of results on Bessel, Macdonald, Hankel, Whittaker, hypergeometric, confluent hypergeometric functions and on different classes of orthogonal polynomials, including polynomials of a discrete variable. The second volume is devoted to properties of special functions which appear in the theory of representations of matrix groups of arbitrary order and to the study of q-analogues of classical special functions which were proved to be connected with representations of Lie groups over finite fields (they are called Chevalley groups). Let us note that interest to q-orthogonal polynomials and basic hypergeometric functions was extended after appearance of quantum groups and quantum algebras. q-orthogonal polynomials with q = ps, where p is a prime number and s is a positive integer, are connected with irreducible representations of Chevalley groups. Namely, zonal spherical functions and intertwining functions of these representations are expressed in terms of these polynomials. By using these groups, addition and product formulas, orthogonality relations and other properties of these q-orthogonal polynomials are proved. It was shown several years ago that q-orthogonal polynomials and basic hypergeometric functions with arbitrary q are related to representations of quantum groups. Different interrelations of the theory xiii

xiv of q-orthogonal polynomials with representations of simplest quantum groups are considered in the third volume of the book [371]. These interrelations are, in principle, such as in the case of classical special functions and representations of Lie groups. However, technical realization of these interrelations is more complicated since the structure of quantum groups is more complicated than that of Lie groups. The third volume deals also with different generalizations of hypergeometric functions, which appear under considerations of matrix elements of representations with respect to the Gel'fand-Tsetlin bases. Special functions with matrix indices are constructed there. The third volume contains also exposition of the theory of special functions of matrix argument. This direction is developed in the present book. The present book contains 6 chapters. Chapter 1 is devoted to Dunkl's results on h-harmonic polynomials and the h-hankel transform. They are related to symmetries with respect to Coxeter groups. Dunkl's differential-difference operators Ti, i =1,..., n, associated to reflection groups are of great importance here. These operators commute with each other. The h-laplacian t::.h is of the form t::.h = Tl +... + T~. The theory of h-harmonic polynomials is an analogue of the theory of usual harmonic polynomials. h-harmonic polynomials are polynomials on the Euclidean space En vanishing under action of the h-laplacian. The analogue of a majority of results from the theory of harmonic polynomials are true for h harmonic polynomials. There exists the operator (which is called the intertwining operator) which allows the transfer of results about ordinary harmonic polynomials onto those related to h-harmonic polynomials. The theory of h-harmonic polynomials leads to the h-bessel function and to the h-hankel transform. Chapter 2 deals with symmetric polynomials and symmetric functions (by symmetric functions we mean here "symmetric polynomials" in infinite number of indeterminates). After description of traditional well-known symmetric polynomials and symmetric functions (monomial symmetric polynomials, elementary symmetric polynomials, complete homogeneous symmetric polynomials and the corresponding symmetric functions), we expose Hall-Littlewood polynomials and functions, Jack symmetric polynomials and functions, Macdonald's symmetric polynomials and functions, and Macdonald's polynomials associated to root systems. All of these polynomials and functions are eigenfunctions of the appropriate second order differential operators. Since these operators are selfadjoint then the corresponding collections of polynomials and functions are orthogonal. These types of orthogonal polynomials are important for different branches of mathematics and mathematical physics. By means of Jack polynomials, a new type of hypergeometric functions is constructed which generalize hypergeometric functions of a matrix argument. Recent papers show that Macdonald's orthogonal polynomials associated to root systems are closely related to zonal spherical functions for quantum groups. Hypergeometric functions related to Jack polynomials are studied in Chapter 3. They are series over partitions of nonnegative integers containing Jack polynomials with certain coefficients as summands. This chapter contains also description

of orthogonal symmetric polynomials associated to Jack polynomials (generalized Jacobi polynomials, generalized Laguerre polynomials, and generalized Chebyshev polynomials), multivariate hypergeometric functions related to root systems, symmetric multivariate Jacobi polynomials associated to root systems, and basic multivariate hypergeometric functions associated to Schur polynomials. There exist various relations between multivariate hypergeometric functions of different types. For example, hypergeometric functions 2F;d) related to Jack polynomials are connected with multivariate hypergeometric functions related to the root system Ben. Clebsch-Gordan coefficients and Racah coefficients of compact groups constitute orthogonal systems of functions in discrete variables. These coefficients are of great importance for mathematical physics and for different branches of physics (nuclear physics, elementary particle theory, atomic physics and so on). As in the case of the group SU(2), they can be used to construct multivariate orthogonal polynomials. Clebsch-Gordan and Racah coefficients of compact groups are studied in detail in Chapter 4. Results of this chapter generalize the corresponding results of Chapter 8 of our book [371], where Clebsch-Gordan and Racah coefficients of the group SU(2) and of some other simplest groups are considered. The interesting phenomenon appears for Clebsch-Gordan coefficients of groups of higher ranks: Clebsch-Gordan coefficients reduce to a sum of products (or to product) of so-called scalar factors (reduced Clebsch-Gordan coefficients) and Clebsch-Gordan coefficients of an appropriate subgroup. Chapter 5 is devoted to studying the denominator function, related to scalar factors of Clebsch-Gordan coefficients of the unitary group U( n) with respect to the subgroup U(n -1). We also expose the results concerning relationship between Racah coefficients, scalar factors and matrix elements of the representation operator Tm(gn-l(71'l2)) of the group U(n). They can be qualified as deepest results in the representation theory of U( n). A certain type of Clebsch-Gordan coefficients of the group U(n) is related to generalized hypergeometric series well-poised in U(n). These hypergeometric functions are also studied in Chapter 5. The theory of these functions generalizes many results from the theory of other types of hypergeometric functions, from combinatorics and so on. Gel'fand's hypergeometric series and functions are described in Chapter 6. They are different types of Horn multivariate hypergeometric series and functions adapted to lattices. There are generalizations of these functions called (~, V) hypergeometric functions. These generalizations contain, as special cases, multivariate q-hypergeometric functions and difference analogues of classical hypergeometric functions. At the end of this chapter, hypergeometric series and functions on Grassmannian G3,6 and on its strata are considered in detail. We see that hypergeometric functions on G3,6 and on strata lead to many well-known types of hypergeometric functions. We often refer to our previous book [371]. Hence, the present exposition strongly depends on the material of that book. We assume that the reader is well familiar with first three chapters in [371]. xv

xvi During preparation of this book the irreparable disaster happened. At the end of 1991, Prof. N. Ja. Vilenkin died. Preparation of this book was finished by the second author. Let us note that the book" Algebraic Structures and Operator Calculus" by Ph. Feinsilver and R. Schott, dealing with special functions and their applications, is under publication. The first volume titled as "Representations and Probability Theory" is already edited by Kluwer Academic Publishers. The second volume "Special Functions and Computer Science" will be published soon. The book "Hypergeometric Functions and the Representation Theory of Lie Algebras and Quantum Groups" by A. Varchenko, dealing with connection of multivariate generalizations of hypergeometric functions with representations of classical and quantum affine Lie algebras, will be published by World Scientific Publishing Company (Singapore). Our book actually has no intersections with those books.